By Allen Hatcher

ISBN-10: 052179160X

ISBN-13: 9780521791601

ISBN-10: 0521795400

ISBN-13: 9780521795401

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1. 3. n C 1/. n C 1/. 2. 2. n C 1/2 . 3 illustrates how 3 times the sum of consecutive oblong numbers is the product of three consecutive numbers. k C 1/to the bottom, then the left, and then the back side of the existing stack of cubes [Kung, 1989]. 3. 1 on 32 CHAPTER 2. Enumeration page 1. When bn denotes the number of blocks in level n, it is easy to show that bn D On 1 C 1 by rearranging the blocks in each layer. 4b for odd n (here 7). 4. On 1 C 1/ D C 24 D 4624: T D nD1 nD1 3 For additional ways to count the blocks in the sculpture, see [Koehler, 2013].

9. 4) 4 We can illustrate this with unit cubes as follows (the illustration is for n D 4/. 10a. 4. 10. 10b, which proves the result [Cupillari, 1989; Lushbaugh, 1965]. 1 shows how cannonballs were stored at Fort Monroe in Hampton Roads, Virginia, a Union military installation, in 1861. The piles of cannonballs had both triangular and rectangular bases. 1. How many cannonballs are there in a pyramidal pile? 2a we see cannonballs in a pile whose shape is that of a pyramid with a square 36 CHAPTER 2.

A modern definition of a pyramid would be that it is a polyhedron with one face (the base) a polygon and the other faces triangles with a common vertex. 17. 17. 20 CHAPTER 1. 18a) is home to much speculation and some surprising coincidences. Some pyramid enthusiasts claim that the Great Pyramid was built so that the area of each triangular face is equal to the square of the height of the pyramid. b=2/2 . For convenience set b D 2 so that s D h2 and s 2 D h2 C 1, thus s 2 D s C 1. Hence s is the golden ratio p ' 1:618 and h D ' 1:272.

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